SECOND ORDER STATISTICS AND CHANNEL SPECTRAL EFFICIENCY FOR SELECTION DIVERSITY RECEIVERS IN WEIBULL FADING Nikos C. Sagias', George K. Karagiannidis2, Dimitris A. Zogas3, P. Takis Mathiopoulos2, George S. Tombras', Fotini-Niovi Pavlidou4 'Laboratory of Electronics, Physics Department, University of Athens, Panepistimiopolis, 15784, Athens, Hellas,
[email protected],
[email protected] 21nstitute for Space Appdications and Remote Sensing, National Observatory of Athens, Metaxa & Vas. Pavlou Street, Palea Penteli, 15236, Athens, Hellas, gkarag@,spacc.noa.gr,
[email protected] 3Electrical & Computer, Engineering Department, University of Patras, Rion, 26442, Patras, Hellas,
[email protected] 4Electrical and Computer Engineering Department, Aristotle University of Thessaloniki, GR-54 124, Thessaloniki, Greece,
[email protected]
Abstract - Motivated by the suitability of the Weibull distribution to model multipath fading channels, the second order statistics and the spectral efficiency (SE) of L-branch SC receivers are studied. Deriving a n'ovel closed-form expression for the probability density function (pdf) of the SC output SNR, the average level crossing rate (LCR), the average fade duration (AFD) and the Shannon's average SE, at the output of the SC, are derived in closed-forms. Our results are sufficiently general to handle arbitrary fading parameter and dissimilar branch powers. Keywords - Selection combining (SC)i, Weibull fading channels, level crossing rate (LCR), average fade duration (AFD), spectral efficiency (SE). I. INTRODUCTION
S
ELECTION combining (SC) receivers are utilized to mitigate the detrimental effects of channel fading and co-channel interference in wireless digital communications systems. Their major advantage is the reduced complexity compared to other well-known diversity techniques, such as equal-gain combining (EGC), maximal-ratio combining (MRC) and generalized-selection combing (GSC). In Lbranch selection diversity receivers, the instantaneous signalto-noise ratio ( S N R ) of the L branches are estimated and the one with the highest value is selected [l]. The second order statistics and the cbannel capacity, in Shannon's sense, of SC receivers have been extensively studied in the open technical literature for Rayleigh [2], [3], Rice [4]-[6] and Nakagami-m fading channel models [7]-[9]. Hoever, experimental fading channel measurements show, that the Weibull model also exhibits excellent fit for indoors [lo], [Ill, as well as outdoors [12], [I31 environments. To the best of the authors' knowledge, only a published work related to the second order statistics and the average channel spectral efficiency (SE) has been presented by Sagias et al. in [14]. In this work, novel analytical expressions for the average level crossing rate (LCR), the average fade duration
0-7803-8523-3/04/$20.00 02004 IEEE.
(AFD) [2] and the average SE, when the Weibull model is considered, have been extracted. However, such performance metrics have not been previously addressed for SC receivers in Weibull fading. In this paper, the channel SE and the second order statistics of SC receivers for non-identical Weibull fading channels, are studied. More specifically, deriving a novel and mathematical trackable formula for the probability density function (pdf) of the SC output SNR, closed-form expressions for the average LCR, the AFD and the average SE, at the output of the combiner, are obtained. Selected numerical examples are presented, showing the effects of various channels and systems parameters, such as the fading severity and the number of diversity branches on the SC performance. 11. SYSTEM AND CHANNEL MODEL We
consider
where and K xz,e(t) = f i n e Ck=l s k , e sin (21rf k , e t - 8 k , ( ) are the inphase and quadrature components of a narrow-band process at timing instance t in the lth input branch, C = 1 , 2 , . . . , L. In the above equations, f i a e S k , e is the fading amplitude of the lcth wave which satisfies ~ ~ =S i:, g==, 1, 8 k , e is the random phase uniformly distributed in [ 0 , 2 1 r ) , K: is the total number of waves and f k > e is the Doppler shift given by fk,e = f d c o s ( Q k , e ) , where fd is the maximum Doppler shift and &,( is the corresponding angle of wave arrival. Taking into account the central limit theorem, for fixed t and for a large value of IC, xl,e(t) and ~ 2 , e ( t )can be considered as zero mean Gaussian random variables (rv) with variance U;. Here, it is convenient to alleviate this notation, by omitting the variable t in the equations. It is well-known that a sum of two quadrature Gaussian rvs is also a Gaussian one, i.e., ze = X 1 , e 3 x2,e = xe eJVe, where = -1. The phase pe = tan-' ( ~ 2 , e / ~ l , of e ) this new Gaussian rv is uniformly distributed in [0,27r) and the =
Xl,t(t)
a
2140
an
fiat
L-branch
SC
receiver,
Et=:=, S k , e cos ( 2 . i r f k , e t - # k , e )
+
amplitude xe =
&!>e +
is Rayleigh distributed, having
Pdf pze (.e) = 2 xe exp ( - x $ / O e ) / R e
(1)
The cdfs of r and ysc are the probability that the signal levels of all branches fall below a certain level, which using (4) and (8), can be expressed as
can be easily obtained as p,, (re) = pe T 2 - l exp ( - @ / f i e )
/Re
(3)
n L
02,
where Re = E ( x z ) = 2 with E (.) denotes expectation. Let the received Sam led envelope in the Ah diversity branch be ye = zi”e = x i A e3 q e / f l e , where ,Be is a positive real constant value. Using (l), the pdf of the envelope
F,.(r)
=
[I - exp ( - r P k / f l k ) ]
(11)
k= 1
and
respectively. The pdf of ysc is obtained by differentiating (12) with respect to ysc,yielding
0
with E r? = Re. It is easily recognized that (3) follows the Weibull istribution [15]. The parameter pe is the Weibull fading parameter (@e 2 0) and expresses the severity of fading. As the value of increases, the severity of the fading decreases, while for the special case of = 2, (3) reduces to the well-known Rayleigh pdf. The corresponding cumulative density function (cdf) and the nth moment of re are
FTc(re)= 1 - exp ( -rfle/Oe e
1
(4)
and
r (d,,e)
E (re“)=
(5)
(13) The above expression for the pdf of ysc can not be easily manipulated in the current form. Therefore, we rearrange (13), performing all the multiplications required and thus, for ,& = ,8, b’e (ae = a and dn,e = d n ) , valid for practical applications, after manipulations (13) can be efficiently written as
respectively, where (.) is the Gamma function [16, eq. (8.310/1)] and d,,e is defined as dn,e = 1 n/,&. Let the instantaneous input SNR per symbol in the &h branch be as
L
+
k=l L-k+l
Ye = re2 E,/No
L-k+2
L
L
(14)
(6)
and the corresponding average input SNR per symbol as -
Te =
(7)
Es/No
3
where E, is the fading -is the transmitted symbols’ energy, power, T; = R2/Pe r (d2,e) and No is the single-sided noise power spectral density (psd) of the additive white Gaussian noise (AWGN), assumed identical and uncorrelated among the L diversity branches. Using (9,(6) and (7) with (4), the cdf of ye can be written as
where ae = l/r (d2,e). The envelope r at the output of the SC receiver will be the one with the highest value among the L branches, i.e.,
r = max { r e }
where
(10)
(L;l)
= ( L - l)!/[k! ( L - 1 - k ) ! ] .
111. AVERAGE FADEDURATION (AFD) AND LEVEL
(9)
and having assumed identical noise psd to all diversity branches during the sampling process, the instantaneous output SNR will be ysc= max { r e } = r2 E,/No.
where ue = p/ (aTe)p/2 and te = exp ( U e $ L 2 / p ) , respectively. Equation (14) includes only sums of simple products of powers and exponential functions, which are mathematically trackable. For independent and identically distributed (i.i.d.) input paths (Ti = 5 and Re = R , V Q using the binomial identity [16, eq. (1.1 ll)], (13) can be simplified as [17, eq. (4)]
CROSSING RATE(LCR) The average LCR and the AFD are two criteria which statistically characterize the fading communication channel. The average LCR is defined as the average number of times per unit duration that the envelope of a fading channel
21 41
crosses a given value in the negative di:rection. The AFD corresponds to the average length of time the envelope remains under this value once it crosses it in the negative direction. Both reflect the correlation properties, and thus the second-order statistics, of the fading channel and they provide a dynamic representation of the channel [2]. The average LCR and the AFD are defned as
N ( r )= and T(7-1
La
i.pp,, (?, r ) df
(16)
c-(r)/iv(T-)
J$=l
kfi
F T k ( r )and by replacing, together with (3), (4) and
(19) into (25), the average LCR of the SC operating in Weibull fading can be obtained in closed-form as
(26) Now, replacing (1 1) and (26) into (17), the AFD of the SC is also expressed in closed-form as
(17)
respectively, where p+,, (i.,r) is the joint pdf of r and its time derivative i.. Using (2), the time derivative of re is .”, .
where j.e is the time derivative of xe. For isotropic scattering, j.e is Gaussian distributed with zero mean and variance [2]
&-e“
fd”.
= a; 2 2
(27) For i.i.d. input paths, after normalizing the signals’ levels to its root mean square (rms) value p = r/rrms,with T, = (26) and (27) reduce to
G,
(19)
From (18), the pdf of i.e conditioned on re is also a zero mean Gaussian distribution, with standard deviation
erg= 2 ,:-Pel2
delbe.
(20)
and
Using (9), the time derivative of the envelope at the output of the SC receiver is . . r = ri, r, = max{re} . (21) From (18) and (21), it can be easily recognized that i. conditioned on the re is a zero mean Gaussian rv, with if (ri = max {re} 1 ri = r ) and pdf variance 6: = pi- (?IT)
= exp (-0.5i.2/d:t)
Consequently, 5, is
a
/
(&C?,~
1.
(22)
discrete rv with pclf
L
P (8,= d2)b (8,- 6%)
p e r (6,)= 2=1
L
P (rz= max {re) I r, == r ) 6 (6,
=
- dz).
2=1
(23) Using (16) and (22) and taking into account p + , , ( f , r )= ( T ) , the average LCR conditional on 8, is
pc ( ? l r ) p ,
N (rid,) = p , ( r ) d,/dG.
(24)
By averaging (24) over the pdf of ti,, i.e., N ( r ) = N (rid,) pe7 (8,) dd, and using (20) and (23), yields
Talung into account the independence assumption between the input branches, l‘ (ri = max {re} l ri = r ) =
T(P)
=
L f d 6 (P/v@’2
exP
[- (P/&)”]
(29)
respectively. Note, that when ,8 = 2, (26) and (27) reduce to previously published expressions for the average LCR and AFD of the well-known Rayleigh model [9]. For L = 1, (28) and (29) can be further reduced to [14, eq. (12)] and [ 14, eq. (1 3)], respectively. The value which maximizes the average LCR can be derived as aN(r)/arIr=,,,, = 0. For L = l, it can be shown that the average LCR is maximized at pmax = 2T1/P& as N (Prnax) = f d m , where pmaa: = rmaa:/r,ms . ~t is interesting to note that the severity of fading does not affect N(pmax). However, for L > 1, ,omax can not be analytically extracted and any of the well-known software mathematical packages such Mathematica and Maple can be used for numerical evaluation. Using (28) and (29), Figs. 1 and 2 plot the normalized average LCR and AFD, respectively, for a dual branch SC with i.i.d. input branches SNRs as a function of p, for several values of p. As the fading severity increases, the normalized average LCR increases, meaning that fades occur more frequently. Furthermore, the lower the signal levels are, the less frequently that they are crossed, whereas higher signal level are crossed more frequently. Furthermore, in Figs. 3 and 4, the normalized average LCR and AFD, respectively, for an L-branch SC with i.i.d. input branches SNRs, is plotted as a function of p for ,8 = 2.5 and different diversity orders. As L increases, the frequency at which the
2142
1o2
IO
5 9
IO‘
1”
-17.5 -15.0 -11.5 -10.0 -7.5
-5.0
-2.5
0.0
2.5
5.0
7.5
-10.0
-7.5
-5.0
20 log,,@)(dJ3
-2.5
0.0
2.5
5.0
7.5
2010g,,@) (dJ3
Fig. 1 Normalized average LCR vs. normalized envelope level for L = 2.
F h n d i z e d AFD
received simal crosses high values (e.g. P > 2.5dB) stays almost the same. Moreover, as L increases fades occur rarely.
special case of ,5’ being integer, k = 2 and 1 = ,B. For i.d.d. input paths (31) reduces to (32) (top of the next page) and setting L = 1 in (32), leads to [ 14, eq. (17)].
VS.
Fig. 2 n0mdized envelope level for L = 2 .
IV. AVERAGE SPECTRAL EFFICIENCY (SE)
The SE, in Shannon’s sense, is an important performance measure since it provides the maximum transmission rate which can be succeeded, in order to the errors be recoverable. The average SE is expressed as [3]
SE
=
CO
log2 (1 + 7)PYsc (7) d7.
(30)
By replacing (14) into (30), it is required to evaluate integrals of the form ~ f i / ~ ln(1 - ’ + x) exp ( - < ~ f i / ~dx. ) This type of integral has been analytically solved in [14], using [lS] and the average SE can be obtained in a closed-form expression as in (31) (top of this page), where G[.]is the tabulated Meijer’s G-function [ 16, eq. (9.301)], available in most of the well-known mathematical software packages, such Maple and Mathernatica and A ( n , t ) = t / n , ( E l)/n, . . . , (e+n- l ) / n , with an arbitrary real value and n a positive integer. Moreover, the values of integers K and p must be chosen, so that p / =~ p/2 holds (e.g. for p = 3.4 we have to choose k = 10 and 1 = 17). Note, that for the
